Infinite Regress

 
Some of the arguments for the existence of God as a First Cause depend on the assertion that there can be no infinite regress of causes.  It is safe to say that although such an assertion is plausible, it is not self-evident. I thought it would be just as well to attempt to construct an example.  My attempt fails, but it is an interesting failure (at least to me).

The idea is this:  Achilles throws his spear, which we will take to be made of continuous, infinitely divisible matter.  (Sorry Democritus!)  In making his through, he obviously does not hold the spearhead directly; he holds the spear by somewhere near the middle of the shaft, but the push of his hand is ultimately responsible for the motion of the spearhead.  Ultimately responsible, but not directly responsible!  We can divide the shaft between his and and the spearhead (of length L) into halves:  Achilles pushes the first half, the first half pushes the second half, and the second half pushes the spearhead.  We can divide this again, so that now we have quarters:   Achilles pushes the first quarter, which pushes the second quarter, which pushes the third quarter, which pushes the fourth quarter, which pushes the spearhead.  We can continue, a la Zeno, to divide the shaft indefinitely, but each time we do, there remains a well-defined chain of causes, with the last piece (ending at distance L) being pushed by the penultimate piece (ending at distance (1 - 2^-N)L).  Isn't this an example of an infinite regress?

Well, no.  It is a finite regress of arbitrary length, and no one has ever had a problem with finite (though long) regress.  We are safe as long as the number of divisions of the shaft is an arbitrarily large but finite number, but what happens if we do what the Greeks would never feel comfortable in doing and "complete" the infinity of divisions? Then the last piece, at distance L, will be pushed by the penultimate piece at distance (the largest rational number less than 1)L.  But there is no such thing as "the largest rational number less than 1".

Of course, another way to avoid a first cause is to have a closed causal loop.  Those may be possible in General Relativity, but it is highly doubtful that they can describe the universe as a whole.  Worse, although the cause of each element in the loop may be well defined, the loop as a whole seems very ... artificial, a thing even more unsatisfying and in need of an explanation than a chain of events going back into the infinite past (with apologies to the Big Bang).